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\newtheorem{case}[theorem]{Case}
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\begin{document}

\title{Quick reference for Johnston, King, and Lie (2014)\\
"Straightforward approximate stochastic equilibria for nonlinear Rational
Expectations models"}
\date{}
\author{}
\maketitle

\section{Model}

Consider a rational expectations model which obeys regularity conditions for
higher order differentiation in which some equations that hold exactly and
some which hold in expectation: 

\begin{eqnarray*}
G\left( z_{t},z_{t+1},\eta _{t+1}\right)  &=&0 \\
E_{t}F\left( z_{t},z_{t+1},\eta _{t+1}\right)  &=&0
\end{eqnarray*}

The solution in terms of differentials is 

\[
z_{t}=dz_{t}+\frac{1}{2}d^{2}z_{t}+\frac{1}{6}d^{3}z_{t}+...
\]%
where the differentials and their laws of motion are outlined below. 

\section{First order state space}

\begin{eqnarray*}
dz_{t+1} &=&\Pi ds_{t+1} \\
ds_{t+1} &=&\Phi _{s}ds_{t}+\Phi _{\eta }\eta _{t+1}
\end{eqnarray*}

\section{Second order state space}

\begin{eqnarray*}
d^{2}z_{t} &=&\theta _{z}^{(2)}+\theta _{z\xi }^{(2)}\tilde{\xi}_{t}^{(2)} \\
\tilde{\xi}_{t}^{(2)} &=&\theta _{\xi }^{(2)}+\theta _{\xi \xi }^{(2)}\tilde{%
\xi}_{t-1}^{(2)}+\theta _{\xi v}^{(2)}\tilde{v}_{t}^{(2)}
\end{eqnarray*}

\begin{eqnarray*}
\tilde{\xi}_{t}^{(2)} &\equiv &\left[ 
\begin{array}{c}
d^{2}s_{t} \\ 
vech(ds_{t}ds_{t}^{T})%
\end{array}%
\right]  \\
\tilde{v}_{t}^{(2)} &\equiv &\left[ 
\begin{array}{c}
vech(\eta _{t}\eta _{t}^{T}) \\ 
vec(ds_{t-1}\eta _{t}^{T})%
\end{array}%
\right] 
\end{eqnarray*}

\section{Third order state space}

\begin{eqnarray*}
d^{3}z_{t} &=&\theta _{z}^{(3)}+\theta _{z\xi }^{(3)}\tilde{\xi}_{t}^{(3)} \\
\tilde{\xi}_{t}^{(3)} &=&\theta _{\xi }^{(3)}+\theta _{\xi \xi }^{(3)}\tilde{%
\xi}_{t-1}^{(3)}+\theta _{\xi v}^{(3)}\tilde{v}_{t}^{(3)}
\end{eqnarray*}%
with%
\begin{eqnarray*}
\tilde{\varsigma}_{t}^{(3)} &=&\left[ 
\begin{array}{c}
ds_{t} \\ 
vec\left( ds_{t}\tilde{\xi}_{t}^{(2)T}\right) 
\end{array}%
\right]  \\
\tilde{v}_{t}^{(3)} &=&\left[ 
\begin{array}{c}
\eta _{t} \\ 
vec\left( \eta _{t}\tilde{\xi}_{t-1}^{(2)T}\right)  \\ 
vec\left( ds_{t-1}\tilde{v}_{t}^{(2)T}\right)  \\ 
vec\left( \eta _{t}\tilde{v}_{t}^{(2)T}\right) 
\end{array}%
\right] \text{ .}
\end{eqnarray*}

\end{document}
